Separating Sources for Encryption and Secret Sharing

Yevgeniy Dodis and Krzysztof Pietrzak and Bartosz Przydatek

Most cryptographic primitives such as encryption, authentication or
secret sharing require randomness. Usually one assumes that perfect
randomness is available, but those primitives might also be realized
under weaker assumptions. In this work we continue the study of
building secure cryptographic primitives from imperfect random
sources initiated by Dodis and Spencer (FOCS'02). Their main result
shows that there exists a (high-entropy) source of randomness
allowing for perfect encryption of a bit, and yet from which one
cannot extract even a single weakly random bit, separating
encryption from extraction. Our main result separates encryption
from 2-out-2 secret sharing (both in the information-theoretic and
in the computational settings): any source which can be used to
achieve one-bit encryption also can be used for 2-out-2 secret
sharing of one bit, but the converse is false, even for high-entropy
sources. Therefore, possibility of extraction strictly implies
encryption, which in turn strictly implies 2-out-2 secret sharing.